1. Introduction
Before adding or subtracting algebraic expressions, it is important to know which terms can be combined. These are called like terms. Terms that cannot be combined are called unlike terms. Understanding this difference makes algebra easier and prevents mistakes.
2. Definitions of Like and Unlike Terms
Like Terms: Terms that have the same variables raised to the same powers. Only the coefficients may differ.
Unlike Terms: Terms that have different variables or different powers. These terms cannot be combined.
Examples:
- Like: \(3x\) and \(7x\)
- Unlike: \(4x\) and \(4y\)
2.1. Characteristics of Like Terms
- Same variable(s)
- Same powers
- Coefficients can be different
2.2. Characteristics of Unlike Terms
- Different variable(s)
- Or different powers on same variable
- Cannot be combined through addition or subtraction
3. Examples of Like Terms
Here are some examples of like terms:
- \(5x\) and \(-3x\)
- \(2a^2\) and \(7a^2\)
- \(4mn\) and \(9mn\)
3.1. Table of Like Terms
| Term 1 | Term 2 | Like? | Reason |
|---|---|---|---|
| \(3x\) | \(9x\) | Yes | Same variable \(x\) |
| \(4y^2\) | \(7y^2\) | Yes | Both have \(y^2\) |
| \(6ab\) | \(-2ab\) | Yes | Same variables \(a, b\) |
4. Examples of Unlike Terms
Here are some examples of unlike terms:
- \(3x\) and \(3y\)
- \(2a^2\) and \(2a\)
- \(5mn\) and \(5m\)
4.1. Table of Unlike Terms
| Term 1 | Term 2 | Like? | Reason |
|---|---|---|---|
| \(3x\) | \(3y\) | No | Different variables |
| \(4a^2\) | \(4a\) | No | Different powers |
| \(6mn\) | \(6m\) | No | Missing variable \(n\) |
5. Why Like Terms Can Be Combined
Like terms can be added or subtracted because they represent the same type of quantity. Only the coefficients change when combining like terms.
For example:
- \(3x + 4x = 7x\)
- \(5a^2 - 2a^2 = 3a^2\)
5.1. Why Unlike Terms Cannot Be Combined
Unlike terms have different variable parts, so they represent different quantities.
For example:
- \(3x + 4y\) cannot be added
- \(5a^2 + 2a\) cannot be simplified
6. Mixed Examples with Step-by-Step Identification
Let’s identify like and unlike terms in these expressions:
6.1. Example Set
Expression: \(5x + 7y - 3x + 9y\)
Like terms:
- \(5x\) and \(-3x\)
- \(7y\) and \(9y\)
Unlike terms:
- Any x-term with a y-term
Expression: \(4a^2 + 3a - a^2 + 6\)
Like terms:
- \(4a^2\) and \(-a^2\)
Unlike terms:
- \(3a\) and \(6\)
7. Quick Practice
Identify the like and unlike terms in the following expressions:
- \(3x + 5x - 7y\)
- \(2a^2 + 4a + a^2\)
- \(5mn - 3m + 2mn\)
- \(9p - 2p + q\)
8. Summary
- Like terms have the same variables with the same powers.
- Unlike terms have different variables or different powers.
- Like terms can be combined by adding or subtracting their coefficients.
- Unlike terms cannot be added or subtracted.
- Identifying like and unlike terms is the foundation of simplifying expressions.